Name: principal strain directions in 3d
Uploaded: 2024-05-23T12:59:55.000Z
Duration: 1 min 29 s
Channel: JoVE

A análise de deformação tridimensional usa os principais eixos de tensão para avaliar a deformação em materiais elásticos e homogêneos.
Um pequeno elemento cúbico, quando expandido em torno desses eixos, se transforma em um paralelepípedo retangular, ilustrando a deformação.
A análise de deformação envolve a rotação de um elemento em torno de um eixo principal, como o eixo n, e a avaliação de componentes de deformação em faces perpendiculares a esse eixo. Este método é baseado na transformação de deformação plana.
Usando o círculo de Mohr, a transformação de deformação é examinada à medida que o cubo gira em torno dos eixos principais. Essa abordagem identifica a deformação máxima de cisalhamento em um determinado ponto, equivalente ao diâmetro do maior dos três círculos.
Na deformação plana, o eixo n se transforma em um eixo principal com deformação zero na origem do diagrama de círculo de Mohr. As deformações principais em lados opostos desta origem representam as deformações normais máximas e mínimas.
Em placas finas de estruturas que sofrem tensão plana, o eixo n se transforma em um eixo de tensão principal.
A deformação principal ao longo do eixo n está associada a deformações no plano. A rotação em torno do eixo m define a deformação máxima de cisalhamento.

análise tridimensional de deformações

Three-Dimensional Analysis of Strain

A análise tridimensional de deformações é crucial para compreender como os materiais se deformam sob tensão, particularmente em materiais elásticos e homogêneos. Este método emprega eixos de tensão principais para simplificar estados de tensão complexos em formas mais simples de compreender. Sujeito a tensões, um pequeno elemento cúbico dentro de um material se expande ou se contrai ao longo desses eixos, transformando-se em um paralelepípedo retangular. Esta transformação ilustra efetivamente a deformação do material. Os principais eixos de tensão são ortogonais, representando direções onde a tensão não induz cisalhamento no material.

O círculo de Mohr é uma ferramenta essencial na análise de deformações. Ele fornece uma representação gráfica dos estados de tensão em um ponto e avalia os componentes de deformação quando o elemento gira em torno de um eixo principal, como o eixo n. Esta análise concentra-se nas transformações de deformações planas, onde as deformações na origem do eixo n são zero, simplificando a determinação das deformações máximas e mínimas representadas em lados opostos do círculo de Mohr.

Em cenários como placas finas sob um estado plano de tensão, o eixo n torna-se o eixo de tensão principal. Aqui, a deformação principal ao longo do eixo n se correlaciona diretamente com as deformações no plano do material. A rotação em torno de outro eixo principal, como o eixo m, ajuda a identificar os locais e magnitudes da tensão de cisalhamento máxima, que são cruciais para prever o comportamento do material sob carga e garantir a segurança e a confiabilidade dos projetos estruturais.

Análise Tridimensional de Deformações

A análise de deformação tridimensional usa os principais eixos de tensão para avaliar a deformação em materiais elásticos e homogêneos.
Um pequeno elemento cúbico, quando ...

Education

JoVE Core

Mechanical Engineering

Engenharia Mecânica

Unit 1

Unidade 1

Transformations of Stress and Strain

Transformações de Tensão e Deformação

KEY TERMS AND CONCEPTS

TERMOS-CHAVE E CONCEITOS

principal strain directions in 3d

Three-dimensional strain analysis shows how a material changes shape when it is loaded. It is especially useful for elastic, homogeneous materials. A small cubic element inside the material can expand or contract along principal stress axes. That change turns the cube into a rectangular parallelepiped, which makes the deformation easier to picture.

The principal stress axes are orthogonal, or at right angles to each other. Along these directions, the stress does not create shear inside the material. This simplifies a complex stress state into a form that is easier to analyze and compare.

Mohr's circle is a key graph used in strain analysis. It shows the stress state at a point and helps evaluate strain components as the element rotates about a principal axis, such as the n-axis. In plane strain transformations, the strain at the origin of the n-axis is zero. This makes it easier to find the maximum and minimum strains, which appear on opposite sides of Mohr's circle.

For thin plates under plane stress, the n-axis becomes a principal stress axis. In that case, the principal strain along the n-axis is directly related to the in-plane strains of the material. Rotation about another principal axis, such as the m-axis, helps locate the maximum shearing strain and its magnitude. These results are important for predicting how materials behave under load and for supporting safe, reliable structural design.

Principal Strain Directions in 3D

transformation of plane stress

Plane Stress Changes with Rotation

Plane stress changes with rotation in a material element. The analysis looks at a prismatic element inside a cube under plane stress. As the element turns, the normal stress and shearing stress on its faces change in both size and direction. These changes are described with trigonometric functions of the rotation angle. The forces on the rotated faces are compared with the forces on the original faces that are perpendicular to each principal axis. The equilibrium equations use only the forces...

principal stresses

Stress Circle for Principal and Shear Stress

The stress circle shows how normal stress and shearing stress change with angle on a transformed plane. Its center, C, lies on the vertical axis and represents the average normal stress. The radius shows the amount of stress variation as the plane rotates. Points A and B, where the circle crosses the horizontal axis, mark the maximum and minimum normal stresses. At these points, shearing stress is zero. These points define the principal stress planes, where only normal stress, called principal...

principal stresses: problem solving

Finding Principal Planes and Stress Limits

Principal planes and principal stresses are found by analyzing two planes that meet at right angles under shearing, tensile, and compressive stresses. The goal is to use the given stress values to describe how the material is loaded and to locate the planes where the stress state changes. To find the principal planes, use the formula that relates the plane angle to twice the shearing stress divided by the difference between tensile and compressive stresses. After substituting the known stress...

mohr&#39;s circle for plane stress

Reading Stress Components with Mohr's Circle

Mohr's circle is a graphical way to read the stress state at a point in a material under plane stress. It turns normal stress and shearing stress into a two-dimensional picture. This makes stress transformation easier to analyze. The method uses Cartesian coordinates. The horizontal axis shows normal stress, written as σ, and the vertical axis shows shearing stress, written as τ. Two points, A and B, are plotted from the stresses on the element. Point A is at (σx, -τxy), and point B is at (σx,...

general state of stress

Stress Tensor and Principal Stresses

The stress tensor describes the internal forces inside a material. It is used when a material is under external force or deformation. At a point such as O, the tensor can show the stress components on any plane through that point. A tetrahedral element helps show how these forces act. One face, XYZ, is perpendicular to the line OA. The other faces line up with the coordinate axes, and point O is the origin. The stress on the tetrahedron is split into normal and shear components on each face. <br...

yield criteria for ductile materials under plane stress

yield-criteria-for-ductile-materials-under-plane-stress

Tresca and Von Mises Yield Criteria

Ductile materials can yield when they are loaded beyond a safe stress level. Engineers use yield criteria to check whether structural elements and machine parts can handle the expected load without permanent deformation. The yield point is first found with a tensile test, which measures how the material responds to uniaxial stress. A tensile test is not enough when a part faces biaxial or plane stress conditions. In those cases, more advanced criteria are needed to predict failure. One common...

transformation of plane strain

Plane Strain Under Rotated Axes

Plane strain changes when the coordinate axes are rotated. This matters for long bars and similar elongated structures that are loaded uniformly on their sides. In these members, one dimension is much larger than the others, so deformation is mainly studied in a single plane. At a point such as O, the key strain components lie along the x and y axes. The strain along the z-axis is negligible in plane strain. Together, the x and y strain components describe the deformation state in the xy-plane.

mohr&#39;s circle for plane strain

Principal and Shear Strains on Mohr’s Circle

Mohr’s circle for plane strain shows how normal strain and shear strain change on a graph. The horizontal axis is normal strain, written as ε, and the vertical axis is shear strain, written as γ. It uses two plotted points, X and Y, to represent the strain state. Point X has coordinates (εx, -γXY), and point Y has coordinates (εY, γXY). The center of the circle, O, gives the average normal strain. The radius comes from the relationship between normal strain and shear strain, so the circle...

three-dimensional analysis of strain

Three-dimensional strain analysis shows how a material changes shape when it is loaded. It is especially useful for elastic, homogeneous materials. A small cubic element inside the material can expand or contract along principal stress axes. That change turns the cube into a rectangular parallelepiped, which makes the deformation easier to picture. The principal stress axes are orthogonal, or at right angles to each other. Along these directions, the stress does not create shear inside the...

measurements of strain

How Strain Gauges Measure Deformation

Strain describes how a material deforms when a force is applied. Normal strain compares the change in length with the original length. Engineers use this measure to track stretching or compression in materials. For higher accuracy, they often use electrical strain gauges. A strain gauge has a conductive wire on a paper backing that sticks to the surface of the material. It works through the piezoresistive effect, which means the wire’s electrical resistance changes when the material is bent,...

key terms and concepts

Learn how principal stress axes, Mohr's circle, and plane stress help show 3D strain, including maximum, minimum, and shearing strain in materials.

Principal Strain Directions in 3D | JoVE K12

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mechanical engineering

Principal Strain Directions in 3D

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